1

Min-Cost Live Webcast under Joint Pricing of Data, Congestion and

Virtualized ServersRui Zhu1, Di Niu1, Baochun Li2

1Department of Electrical and Computer Engineering

University of Alberta2Department of Electrical and Computer

EngineeringUniversity of Toronto

2

Roadmap

Part 1 A joint pricing of data, congestion and virtualized servers

Part 2 Min-cost multicast as k-NWST

The first PTAS proposed

Part 3 Trace-driven simulations

Part 1 A joint pricing of data, congestion and virtualized servers

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Live Webcast

Problem: Large amount of data transferringSignificantly contributing to traffic congestionEngaging many server resources, etc.

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Charge end users – conventional

Monthly flat rate/ Pay-as-you-go/Both

Excessive burden on clients

Charge content/application provider

Encourage customers to use more

E.g. Telus: free six-month subscription of Rdio

Existing pricing policies

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How should webcast operators pay for the

video delivery service?

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A road pricing motivation

Distance traveled pricing

Transferring data

Congestion specific pricing

Congestion degree

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Congestion pricingCharge the webcast provider

A per-minute price rate on each link

Pricing rate ∝ bandwidth-delay product

Related with the media streaming topology

Encourage webcast operator minimize its “waiting data”

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Cost of servers

Download from

source

Recoding and resendingClient

Operation cost

9

Roadmap

Part 1 A joint pricing of data, congestion and virtualized servers

Part 2 Min-cost multicast as k-NWST

The first PTAS proposed

Part 3 Trace-driven simulations

10

System model

SourceCDN ServersClient

11

F

F

S

F

F

Objective: minimize the total cost including data transferring, congestion

and server opening

12

Formulating the problem

, ,, ,

,

min e e i i i j i jx y z

e E i F i F j T

c z f y c x

( )

,

,

,

, ( )

= 1 ( )

, ( , )

,

, , {0,1} ( , , )

e ie N

i ji F

i j i

ii F

i j i e

z y i N F

x j T

x y i F j T

y k

x y z i F j T e E

，

s.t.Server congestionService congestion

13

Formulating the problem

, ,, ,

,

min e e i i i j i jx y z

e E i F i F j T

c z f y c x

( )

,

,

,

, ( )

= 1 ( )

, ( , )

,

, , {0,1} ( , , )

e ie N

i ji F

i j i

ii F

i j i e

z y i N F

x j T

x y i F j T

y k

x y z i F j T e E

，

s.t.Opening cost

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Formulating the problem

, ,, ,

,

min e e i i i j i jx y z

e E i F i F j T

c z f y c x

( )

,

,

,

, ( )

= 1 ( )

, ( , )

,

, , {0,1} ( , , )

e ie N

i ji F

i j i

ii F

i j i e

z y i N F

x j T

x y i F j T

y k

x y z i F j T e E

，

s.t.

Optimal solution is a tree

Each client belongs to one server

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The data costThe total data transferred per unit time is proportional to the total number of selected edges

Given the video bit rate r, the total data transferred is

Since nr is a constant, this cost can be incorporated into the server opening cost

ii F

y r nr

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Unfortunately, it is a hard problem.

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Let’s start by ignoring the opening costThen, fi=0 for all relay servers.

Only congestion cost are considered.

Equivalent with an very famous hard problem, Steiner Tree. (NP-hard, even within 1.0105)

M. Chlebik, J. Chlebikova.The Steiner Tree problem on graphs: Inapproximability results. Theoretical Computer Science, 2008

If we don’t consider the inter-server connection

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Case 1: No cost for inter-server connections.

Case 2: No inter-server connections are permitted.

In both case, they are equivalent with Uncapacitated Facility Location problem, another NP-hard problem.

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No server number constraint?

Well, it is called Node-Weighted Steiner Tree problem (NWST).

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NWST – Existing Results

NP-hard to approximate withinC.Lund, M. YannakakisOn the hardness of approximating minimization problems. Journal of the ACM, 1994

Currently best known ratio:S. Guha, S. Khuller.Improved methods for approximating node weighted Steiner trees and connected dominating sets. Information and Computation, 1999

(1 ) ln n

1.35ln n

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The linear relaxation, ,

, ,,

min e e i i i j i jx y z

e E i F i F j T

c z f y c x

( )

,

,

,

, ( )

= 1 ( )

, ( , )

,

, , 0 ( , , )

e ie N

i ji F

i j i

ii F

i j i e

z y i N F

x j T

x y i F j T

y k

x y z i F j T e E

，

s.t.

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Original problem

A PTAS for k-NWST

, ,,

min (x, y, z)

s.t. ,

e e i i i j i je E i F i F j T

ii F

f c z f y c x

y k

, ,,

(x, y, z, ) ( )e e i i i j i j ie E i F i F j T i F

L c z f y c x y k

The Lagrangian relaxation

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Lagrange multiplier λ as opening cost: fi’ := fi

+ λ

Subroutine Algorithm1:

A PTAS for NWST with additional opening cost

, ,,

, ,,

(x, y, z, ) ( )

( )

e e i i i j i j ie E i F i F j T i F

e e i i i j i je E i F i F j T

L c z f y c x y k

c z f y c x k

( , , )P c f A G1P. Klein, R. Ravi.A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithm, 1995

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A PTAS for our problemSearching for proper Lagrange multiplier λ 1

1 1

2 2

( )

( )

P

P

A

A

1 1 2 2k k k

Convex combination of P1 and P2

2

If μ2>1/2, output P2. Otherwise, select some nodes in P2 and add them in P1

3

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Step 1: find proper λ

• For sufficiently large λ, the opening cost dominates

• For sufficiently small λ, the cost depends on congestion, making more to open

• The binary search can find two trees near the server constraint

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Step 2: Convex combination

• Convex combination of P1 and P2

1 1 1

2 2 2

( ) ( ) ( )

( ) ( ) ( )OPEN D

OPEN D

C X C X k k OPT

C X C X k k OPT

where is the total opening cost

is the total congestion cost

( )OPENC X

( )DC X

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Step 3: Merge P1 and P2

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Target: select k-k1 nodes from P2

P1

P2

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Double edges of P2

P1

P2

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Find the Euler tour and shortcut to tour

P1

P2

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Find the Euler tour and shortcut to tour

P1

P2Average cost:

Then, we have:1 2 1( ) / ( )k k k k

12 2

2 1

2 2 2

( ) ( ')

2( ( ) ( ))

2 ( ( ) ( ))

OPEN D

OPEN D

OPEN D

C X C P

k kC X C P

k k

C X C P

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Connect P1 to the cheapest path of tour

P1

P2

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The total server cost

1 1

1 1 1

2 2 2

( ( , ')) ( ) ( )

( ') ( ')

( ( , ')) 2 ( ( ) ( ))

2 ( ( ) ( ))

D OPEN D

OPEN D

D OPEN D

OPEN D

C PATH S X C X C P

C X C P

C PATH S X C X C P

C X C P

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The upper bound for total cost

Since , we have1( ( , ))DC PATH P S OPT

1 1 1 2 2 2

1

1 2

( ) ( )

2 ( ( ) ( )) 2 ( ( ) ( ))

( ( , ))

2 2

2

(2 1)

OPEN D

OPEN D OPEN D

DS

C X C X

C X C X C X C X

C PATH P S

OPT OPT OPT

OPT OPT

OPT

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Conclusion (Approximation Ratio)Our PTAS can approximate k-NWST with a ratio of

2 1.35ln 1 2.7 ln 1.n n

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Roadmap

Part 1 A joint pricing of data, congestion and virtualized servers

Part 2 Min-cost multicast as k-NWST

The first PTAS proposed

Part 3 Trace-driven simulations

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Inter-server and server-client delay traces

Traces collected from PlanetLab and from the Seattle projectMonitor the RTTs among 8 Planet nodes for a 15-day periodMonitor the RTTs from the 8 Planet nodes to 19 Seattle nodes

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Opening cost assignment

The opening costs (including data) for CDN edge nodes are from pricing policy by Amazon Web Service (Amazon CloudFront)

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Baseline Algorithm

Randomly chooses a subset of servers to openWith no inter-server connectionsConnects each client to its closet server.

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12345678 0

0.5

1

1.5

2

2.5

3 Total Cost Congestion Cost Opening CostPe

rform

ance

Rati

o

The cost computed by our algorithm

Number of Servers

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12345678 0

0.5

1

1.5

2

2.5

3 Total Cost Congestion Cost Opening CostPe

rform

ance

Rati

o

The cost computed by baseline algorithm

Number of Servers

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ConclusionsA joint pricing policy of data, congestion and virtual servers for live webcasting application providers

Model the Min-cost multicast and provide the first PTAS for it

Future work:

Only routing are considered, how about using network coding?

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Thank you

Rui Zhu

Department of Electrical and Computer Engineering

University of Toronto